Superconnection currents and complex immersions. (English) Zbl 0696.58006

The present paper contains complete proofs of the results which have been announced in C. R. Acad. Sci., Paris, Sér. I 307, No.10, 523-526 (1988; Zbl 0652.32020). Given an immersion \(M'\to M\) of complex manifolds, a vector bundle \(\eta\) on \(M'\), and a finite complex (\(\xi\),v) of Hermitian vector bundles on M which provides a projective resolution of the sheaf of sections of \(\eta\) the author transfers Quillen’s family \((\omega_ u)_{u\in {\mathbb{R}}_+}\) of superconnection currents from \({\mathbb{Z}}_ 2\)- graded bundles to (\(\xi\),v) and investigates the limit \(\omega_{\infty}\). He proves its existence and expresses \(\omega_{\infty}\) in terms of integrals of Gaussian shaped differential forms on the normal bundle N of \(M'\). Under certain compatibility assumptions for the metrics on \(\xi\), N, and \(\eta\) the limit \(\omega_{\infty}\) is explicitly calculated using Chern-Weil representatives of \(Td^{-1}(N)ch(\eta)\). For later applications to intersection theory (Bismut, Gillet, Soulé; to appear) the speed of convergence \(\omega_ u\to \omega_{\infty}\) and the behaviour of the wave front sets are precisely controlled. This together with complicated algebraic identities make the present paper quite technical.
Reviewer: K.Lamotke


58A50 Supermanifolds and graded manifolds
53C05 Connections (general theory)
32L10 Sheaves and cohomology of sections of holomorphic vector bundles, general results
57R42 Immersions in differential topology
57R20 Characteristic classes and numbers in differential topology
53C65 Integral geometry
58A10 Differential forms in global analysis


Zbl 0652.32020
Full Text: DOI EuDML


[1] Atiyah, M.F., Bott, R.: The moment map and equivariant cohomology. Topology 23, 1–28 (1984) · Zbl 0521.58025 · doi:10.1016/0040-9383(84)90021-1
[2] Atiyah, M.F., Singer, I.M.: The Index of elliptic operators. IV Ann. Math. 93, 119–138 (1971) · Zbl 0212.28603 · doi:10.2307/1970756
[3] Berline, N., Vergne, M.: Zeros d’un champ de vecteurs et classes caractéristiques équiva-riantes. Duke math. J. 50, 539–549 (1983) · Zbl 0515.58007 · doi:10.1215/S0012-7094-83-05024-X
[4] Berline, N., Vergne, M.: A proof of Bismut local index theorem for families of Dirac operators. Topology 26, 435–463 (1987) · Zbl 0636.58030 · doi:10.1016/0040-9383(87)90041-3
[5] Bismut, J.M.: The Atiyah-Singer Index Theorem for families of Dirac operators: two heat equation proofs. Invent. Math. 83, 91–151 (1986) · Zbl 0592.58047 · doi:10.1007/BF01388755
[6] Bismut, J.M.: Localisation du caractère de Chern en géométrie complexe et superconnexions. C.R.A.S.-307, Serie I, 523–526 (1988) · Zbl 0652.32020
[7] Bismut, J.M., Bost, J.B.: Fibre determinant, métriques de Quillen et dégénérescence des courbes. Acta Math. (to appear)
[8] Bismut, J.M., Gillet, H., Soulé, C.: Analytic torsion and holomorphic determinant bundles. I. Comm. Math. Phys. 115, 49–78 (1988) · Zbl 0651.32017 · doi:10.1007/BF01238853
[9] Bismut, J.M., Gillet, H., Soulé, C.: Analytic torsion and holomorphic determinant bundles. II. Comm. Math. Phys. 115, 79–126 (1988) · Zbl 0651.32017 · doi:10.1007/BF01238854
[10] Bismut, J.M., Gillet, H., Soulé, C.: Analytic torsion and holomorphic determinant bundles. III. Comm. Math. Phys. 115, 301–351 (1988) · Zbl 0651.32017 · doi:10.1007/BF01466774
[11] Bismut, J.M., Gillet, H., Soulé, C.: Bott-Chern currents and complex immersions. To appear in Duke Math. Journal · Zbl 0697.58005
[12] Bismut, J.M., Gillet, H., Soulé, C.: Complex immersions and Arakelov geometry. (To appear) · Zbl 0744.14015
[13] Bott, R., Chern, S.S.: Hermitian vector bundles and the equidistribution of the zeros of their holomorphic sections. Acta Math. 114, 71–112 (1968) · Zbl 0148.31906 · doi:10.1007/BF02391818
[14] Donaldson, S.: Anti-self-dual Yand-Mills connections over complex algebraic surfaces and stable vector bundles. Proc. London Math. Soc. 50, 1–26 (1985) · Zbl 0547.53019 · doi:10.1112/plms/s3-50.1.1
[15] Eilenberg, S.: Homological dimension and local syzygies. Ann. Math. 64, 328–336 (1956) · Zbl 0073.26003 · doi:10.2307/1969977
[16] Hörmander, L.: The analysis of linear partial differential operators. Vol. I. Grundl. der Math. Wiss. Band 256. Berlin Heidelberg New York: Springer 1983 · Zbl 0521.35002
[17] Manin, Y.: Gauge field theory and complex geometry. Grundl. der Math. Wiss. Band 289. Berlin Heidelberg New York: Springer 1988 · Zbl 0641.53001
[18] Mathai, V., Quillen, D.: Superconnections, Thom classes and equivariant differential forms. Topology 25, 85–110 (1986) · Zbl 0592.55015 · doi:10.1016/0040-9383(86)90007-8
[19] Quillen, D.: Superconnections and the Chern character. Topology 24, 89–95 (1985) · Zbl 0569.58030
[20] Serre, J.P.: Algèbre locale. Multiplicités. (Lect. Notes Math., Vol. 11) Berlin Heidelberg New York: Springer 1965 · Zbl 0142.28603
[21] Berthelot, P., Grothendieck, A., Illusie, L.: Theorie des intersections et theorem de Riemann-Roch. (Lect. Notes Math., Vol. 225) Berlin Heidelberg New York: Springer 1971 · Zbl 0218.14001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.